My Windows 8.1 64 bit computer running Mathematica 10.0.0.0 gives the same value 26727055876061626.... for Det[bigMatrix] obtained previously by Udo Krause and David Park Jr. I have also written a program that diagonalizes an integer matrix by row and column reductions. If the matrix is square the product of the diagonal entries is then the determinant. This gives the same answer, so there is little doubt we have the correct value. Mathematica 9.0.0.0 running on the same computer alternates at random between two values for Det[bigMatrix], both incorrect, and one exactly twice the other. These appear to be the values 2.284217140 x 10^9769 and 1.142108570 x 10^9769 mentioned by David Park Jr.

This is an example of what I think is a misdirection in the WRI approach, one that could probably be easily fixed. That is what I call the lack of "hierarchical depth", the ability to calculate things at all levels. This is partly a result of concentrating too much on high powered, top-down routines. If you want to check them with lower level routines it's all hidden and inaccessible. The more basic routines are either missing or not tailored to more basic usage. It's not that I don't think there should be high level "set-piece" routines for common applications, but just that users should have access to lower level approaches whenever possible. Anyway, I always go for "hierarchical depth" and "calculate everything".

In Presentations I have a Student's Linear Equations section that allows step by step manipulation of matrix structures. I never envisioned it for a research problem as discussed in the PDF. Nevertheless, I thought I would experiment. I stumbled into calculating the determinant by calculating the reciprocal of the determinate of the inverse. In a case that was giving discordant and incorrect results with the straight determinant I obtained consistent, and what looks like correct results. Here is the test case:

powersMatrix = 
  DiagonalMatrix[
   10^# & /@ {123, 152, 185, 220, 397, 449, 503, 563, 979, 1059, 1143,
      1229, 1319, 1412}];

basicMatrix = {{39, 73, -86, -75, 24, -67, -87, 9, -65, -98, 60, 85, 59, -65}, {14, 
  75, 84, 63, -26, -15, -41, 98, -57, 31, 72, -99, 10, -18}, {-47, 
  84, -92, -39, 56, -59, -40, -77, -37, -47, 50, 31, -95, -78}, {0, 
  64, -76, -23, -95, 23, -30, 23, -46, -3, -21, 79, 1, 91}, {29, 
  42, -97, -10, 60, -84, 47, 84, -79, -2, 78, 40, 67, 95}, {-30, 31, 
  23, -78, 28, 16, 3, -91, -13, -80, -60, -29, 
  15, -22}, {-4, -10, -74, -17, -89, -48, 
  59, -78, -77, -59, -48, -59, -20, 41}, {72, 68, 35, -83, -63, 
  55, -27, 86, -3, -19, 51, 91, -9, -20}, {9, 95, -39, 21, 64, -61, 
  23, 93, 38, -68, -95, -28, -61, 19}, {-67, 91, 44, -97, -43, 93, 91,
   58, 16, -7, 48, 55, 40, 9}, {-58, -36, -84, -46, 20, 32, 31, 
  11, -56, 38, -53, 91, 49, -86}, {-81, -29, 33, 37, -41, -58, 
  32, -35, -2, 84, 98, -97, 81, 39}, {-35, -64, -97, 13, -26, -79, 49,
   24, -49, -48, -62, 99, -13, 
  20}, {-36, -44, -91, -66, -8, -70, -64, -8, -62, -70, 38, 1, 19, 
  22}}

smallMatrix = {{992, 461, -757, 484, -177, 976, 716, 277, 311, 825, -899, -848, 
  221, -205}, {308, 232, -30, 0, -20, 630, 205, 767, 426, 177, 422, 
  383, -634, -762}, {411, 174, 618, -619, 18, 849, 377, -274, 
  61, -253, -807, -482, -172, -96}, {-693, 356, -365, 375, 586, 298, 
  245, 924, -161, -67, 362, 181, 692, -151}, {306, 995, -206, 878, 
  215, 692, 356, 48, 869, 294, 984, -915, -877, 909}, {-758, 
  939, -481, -855, -110, 5, -360, -603, -680, 491, 707, 
  518, -78, -49}, {725, -306, -975, -832, -254, -517, 
  806, -664, -523, -167, 787, 829, 925, -642}, {-471, 59, 
  259, -953, -655, 374, 633, 266, -524, -580, -21, 309, 
  340, -168}, {-605, -464, 305, 252, 364, 37, 72, -761, 57, 853, 
  600, -109, -718, 884}, {-151, -470, -329, 368, 237, 
  591, -711, -378, -246, -880, 944, -209, -614, 
  456}, {-148, -366, -764, 374, 609, -702, 58, -214, 
  928, -389, -962, -484, 642, -530}, {-249, -336, -395, 74, 
  166, -820, -486, -870, 637, -420, -593, -862, 344, 273}, {-369, 
  489, -185, 684, -791, -190, -748, -616, -404, -693, 
  514, -422, -198, -831}, {-304, 28, 434, -288, 502, 133, 
  172, -31, -753, 208, 847, -542, -777, 763}}

bigMatrix = basicMatrix.powersMatrix + smallMatrix;

Calculating Det[bigMatrix], and displaying to 10 places, I obtain either 2.284217140 x 10^9769 or 1.142108570 x 10^9769. Neither of these are correct. The reciprocal of the determinant of the inverse gives:

1/Det[Inverse[bigMatrix]] // N[#, 10] &
2.672705588*10^9763

This is the same answer one gets by truncating the elements of bigMatrix and taking the determinant.

bigMatrix2 = N[bigMatrix, 10];
Det[bigMatrix2]
2.672705588*10^9763

Assuming that the answer is not wildly dependent on the precision we might have some reason to trust:

1/Det[Inverse[bigMatrix]]

giving:

2672705587606162693200253937085000000000000000000000000000000000000000\
0000000000000000000000000000000000000000000000000000172614614814380187\
3744406782124594156243273664096717669821188008611374471095367329358081\
1535170005519801715998205770608306130331300000000000000000000000326665\
0674246209123094621358364193355738754341202642780914226469897184310042\
0157114851029953246680074718964934943657745920830494832826636087435261\
7232244329970160000000000000255831989239635286692736634985010028677402\
4012807389350606974012543896094548898376815835808147564254591131229479\
8035312000545131772101187120155441325837715621038171133224482158219460\
2532982693404202950234304828591273519394623033546142866398435725600267\
1738567709607306994543770164338888692431690247858526409746755585856658\
0432245043026227702023943994645915902296798180605554418936597776950816\
9551548598163879398297557606709060904450720588126984121362618171487024\
4004961533285763394707226529491064316333114682961215626234595772846118\
5230669164750702872970802215552641237188496973525030404603055314650955\
4203636533693619143649989208865386899609768556098536757213620019264866\
0684229485175512203421947669909539500368930846401329814419566400401697\
9220169564754600909622330816974811125183198201489490613732945651947188\
5162526402192670889903117187567766999561397064059803380096386790552396\
1336979740604158754859679946166257988690095778204320090101452297792074\
2074618640547070743308188662967776509955727602059893926407614125482022\
0607185409771171015502105996039499131956783496507487988436030908463457\
6088847115120167001705469161062741273884219041284849430551382696380676\
3443652649582683403400225820754304187890284662644860291474064673307934\
3604076510371921112264268843507352921350855058813567741169933593817788\
6181718284991671380078087199867317453420301369645955679171439814313618\
1818754790073027475988223913684279640395370076271071624058688462585446\
1211971182299004751719281949202025134986960478200135644069374120186516\
7031737564051602315254180084128322667926517488354122644960190604633534\
6250160508207334003957275122659277171593771102157191066387536780233220\
8476669612119201659227069595893698322944774535458126162550889842060339\
5733342118949686794845152466056764962162246054692145984677301593012803\
2778862411437059260300890231144715006195015345233009856935102372143717\
9524864947297578990252161606866657671756125476894036757181051200170567\
1501785249452755025006197878951293018537681829796820708977858202296266\
3843274538416839339553185352449689491762727087921772802241821945291996\
9083548224538139751922591252681698727315106604681999203307399118863261\
8767106409959552688190093607634035694746995512006068519689664356309112\
0833231723635686900157516465791063585797849561100560552265615612126509\
6213498662337815787107244897150270670802744446075683796760700950261955\
1962992893836392558095987187341862974361180863954891845189055463764709\
6120628168770332140600534315145068608706814441578895573361630933709645\
2681633988975768756300746485012230561671455031277701666998627921419719\
0316675546119223094497606948999260022493808983282530544574704440665258\
1465008772275420958897250323517324535974903743602262438617749324393901\
1058300002697305355380042461065358965266199531399925371609632492746734\
1511709977639616338814341965171252314759941696766437869267669993256663\
4020506492153455020835521167391940328480334362653676375482289375396049\
0445340741596222516106683680017915504849455567198891167664600196860012\
7353629891225385685034910535448704617845096685396163237932274738020434\
0195847734439239515723172915345707668662624599194154163605444360514034\
4673086296194734513442954830475081165679601422022345110203709820717535\
8852727920415064704619036588359170845775393985192636388982947231911109\
0846938071452757856057512438638290066748655001326861429690651423768338\
5392767500824092819351869960927692073620068607266722381452403994111476\
6723211549043736767741010622607428232834743416370707799251634214522544\
9800721131394512261310698063185838779315500282420222615271434841958001\
4863379099241113772563178729442020837653007589723319093766695830678516\
2219143745925955437338328445739794665365930627263389385179811621133636\
0154922306676636713518505679161390634344890729798727727303390207689122\
1940088411039990838737115970607070162242324251075361488049577438814799\
3315271802038057050087322033061406703975521688477447107623353279153018\
7402139670193198424177958236538734705184050322701648544084556083208587\
1135552623173312351338787872463271244014121083139131568363844148470749\
0049814868213885826295276372300941614401364645848357779953342939096504\
9545395061452657530247825867030521813698179995901662723467807665654813\
3293304929787262393356754514508309593377985270485967382488497059639884\
9198813805418835568287187391024611889351714996208769906627473875029633\
9939781640115811752713710767511480021573812861017649734292097678983443\
9142466012503705553812891534338366699205946210662681457323746249139507\
6607645435061741401733424103634293330175523603725248914259865301413492\
5333147122657732259989368088820481367490144968441881663878197929304616\
7849923201367382228406978313948047597240282356692341479740986581388960\
8540597948196881933737216316672395830514429420309798691738344289718648\
1890228267422523008278311752408175119360170660241954666000996812540544\
7925712110137635012044884912114570125788741250855037878443614319806118\
4786225437297942884167570753082793513889947019334697761455964751633865\
8487108179885522114973300278123330094032232901495345002771109430828373\
0485679061283994395831996917502533327456374363532731172797213263756048\
7346045947621593241719608586645254621551618108994316574792837346805266\
9814220720158895334067798862324807846405704177280316332678905114195363\
6475940728631018390653523864664356274218168664323882644198072000120739\
8721627247534608717384464213095884775945145945893429391987573078754525\
3678014956971556694431190467686223906180991699889437808782448684211457\
0566230695006468388529267576421664189411597190020322627049655527784497\
9072972347949844640981980659424987304899094897884675983729835513536764\
8150813352593435374105201743766477829150923197057688274611573355044386\
0006618872810822831990294942300008823229887480285520652864365013199352\
8070298262656775906327003857880790426131268718509036999336795315748353\
1806038151413444997834686438815749308259886776471671843530003770851147\
8331253057317994586453130505209918391858007488077273584860303197084744\
0519988619719187192435271041678820892412208881211197496271198011935073\
6188830692807734717513447248518996762882053153260227968261907405856561\
5471694962506203081990263094913152759632941388692759852900787635370127\
7891982032051094173630505840693808993568444543076165830374999470482531\
7309760793671580291861056981714983021172104531365929554740359095012607\
9498325261774849018663026549702163138013632259616802317221805613917308\
2380754659845301327274341028656346740589794319090278889796822840434471\
2342744112363268626380136195740740982088243808176812547338081475017899\
0245120819841928304842542061888285597877980345637341953841383063003010\
1178017883159674549295718751961918793943337517968272093722557316678743\
3525173765477589224675162065850467393351083261833876055475431697596122\
6706396487100094570822822107913156638353560488026235276776977362696930\
7318802649809172281522495523148115774674879303213326017492189665459404\
8961655659717438281210121313258117942336179570437892106452997159879264\
6416192603820699115612860027729109082633578603231401410076027883526502\
3898247021664710701493742804630209636015042604178423059116939342584675\
8638516239771162367453552437465310280615694051597062644694956461867446\
7566870081159509132118391213433423380329626605559431904645447505595580\
9450828777173498303817891489386396911977285306347217760300677428028121\
6800288747713355099213078301724777441158434700148416245586870765044383\
6042559948372987791320505500135228978133633692716700370370993506011707\
7575267068683185920738822170750693103731533714364825908792179005142237\
9254362547761382323861590153552296259781620063302526853695682263852507\
1980848788975691428537106576527571991935673077120850576794415438294788\
8828599828021630832834312709140874309139807418823449841598225537904383\
9240309416001235618815134507771400358347730821618745285827289810602391\
4057918646449862311452982223044639631459217745491847379623854095180614\
3823438849583684728187953330697477122137719822986823256260492223460463\
0865582503712647269628284700241907996336618115036794413578008201220432\
7633096451092624022640773355732391421950120131543270135936343919190042\
5939184187663355767187167069119747159059087284982447117781541906458384\
0732525335443413266128667185729842605428024867301876046739944123383676\
2681625581509663603660945080287755799097016649806567545235631750169310\
1605142279133553460727993955277360423698964648116352971060219909220442\
0768617130228037431057066961958520613160306312957533279966102491440300\
4499770827279450047918443367025900978186249906481749466324319072201633\
4157458638447391260979362922627225530149995822506217350199716348992782\
0976981436143788112156945391758333457816340397443665564361600469566510\
5995209075571885257242277817826420496692376865950172240041782130186718\
9634972381589203908584720678978345198588854535399376959832216527085822\
5512593220180437286044149943474143914597182481182493842057351626651723\
5179091500212066112501938424694060652030106711883355699971657979209289\
8896996598004068714977127847253862409102497425915230148643090214488021\
9404986859193306984754661025255771669037402397500434894722185415621673\
3476537432365047803911300619996101726470000000000007171831179178198725\
6753438477942876485294307903666216787369894137378969332590982598060377\
8060197646633025273453610084114853134319482029158379999999999999999999\
9999999999999999999999999999999999999999999999999999999999988579699668\
7284037844559090524468965759586782

Why that would work is a bit of a mystery and maybe it wouldn't always work.

I need to add that this was in Mathematica 9.0.1.

Hi David, I agree with the general approach and attitude outlined in the first paragraph of your post. But as Frank Kampas indicated, there is a certain "triage" perspective that has to be taken when a product like Mathematica is developed.

More precisely, we can say that the "hierarchical approach" you talk about is basically transparency of the algorithms. That transparency of the "high-power routines" can be done in two ways: (i) with API's of sub-routines, as you suggested, and (ii) by detailed documentation with computation examples, comparisons, etc. WRI tries to do both, but WRI's "triage" might produce different outcomes than what ambitious or power users might expect.